1.1 Objectives
This book concerns the modeling of engineering dynamic systems. A model is a representation of an actual system. In this context, first we need to explore what a dynamic system (an engineering system in this case) is; what is specifically meant by modeling; what types of modeling are possible, and how to model a dynamic system. We will address all four topics in detail throughout this book. In brief, a dynamic system is a system where the ârates of changesâ of its response variables (outputs) cannot be neglected. There are many types of engineering dynamic systems and many types of models, as we will learn. This book primarily concerns analytical modeling. We will primarily focus on âlumped-parameterâ models, which depend on the independent variable âtime,â the more general âdistributed-parameterâ models, which have both time and space as the independent variables, are given some attention. We will learn a way to develop an analytical model that has the four characteristics: integrated, unified, unique, and systematic, for an engineering dynamic system. We will explore these four characteristics in detail, in this book.
The main learning objectives of this book are the following:
- Understand the formal meanings of a dynamic system, control system, mechatronic system, and multi-physics (or, multi-domain or mixed) system.
- Recognize different types of models (e.g., physical, analytical, computer, experimental) and their importance, usage, comparative advantages and disadvantages.
- Under analytical models, recognize the general and specific pairs of model categories.
- Learn the concepts of input (excitation), output (response), causality (causeâeffect nature, what are inputs, and what are outputs in the system), and order (dynamic size) in the context of a dynamic system (or dynamic model).
- Understand the concepts of through-variables and across-variables, their physical significance, and relationship to state variables.
- Recognize similarities or analogies among the four physical domains: mechanical, electrical, fluid, and thermal (this is the basis of the âunifiedâ approach to modeling).
- In each physical domain, recognize the lumped elements that store energy and that dissipate energy, based on the analogy among different physical domains.
- In each physical domain, recognize different types of source (input) elements, which possess independent input variables and are able to apply them to other components of a system, based on the analogy among different physical domains.
- Understand the âmechatronicâ approach (i.e., the âintegratedâ or âconcurrentâ approach) to modeling a multi-physics (or multi-domain or mixed) system, which consists of two or more basic physical domains. Integrated means, all domains are modeled (and designed) simultaneously.
- Understand the âunifiedâ approach to modeling a multi-domain system. Unified means, similar (i.e., analogous) methods are used to model the different physical domains in the system.
- Understand the meaning of state variables and the selection of them in a âuniqueâ manner to generate a unique state-space model.
- Learn to apply the unified and integrated approach of modeling, in a systematic way, to develop a âuniqueâ state-space model. Systematic means, the modeling steps are clear and there is no uncertainty associated with it. Unique means, a single model is obtained at the end.
- Understand the key steps of development of a unified, integrated, systematic, and unique approach for modeling an engineering dynamic system. Learn to develop state-space models using that approach, while using physically meaningful state variables that lead to a âuniqueâ state-space model. Also, understand the physical meaning of âsystem orderâ or the dynamic size.
- Learn how to convert a state-space model into an inputâoutput model, in the time domain.
- Learn to obtain a linear model of a nonlinear dynamic system, both analytically and experimentally. In the analytical context, learn different approaches to linearize a nonlinear system or model, particularly the slope-based local linearization and the energy-based global linearization.
- Understand and apply a graphical approach that uses linear graphs, to develop a state-space model.
- Understand the frequency-domain concepts of modeling; particularly, the concepts of âgeneralizedâ impedance, equivalent circuits, and circuit reduction of electrical systems (Thevenin and Norton concepts of equivalent circuits) and transfer function linear graphs (TFLGs) and apply them to mechanical, fluid, thermal, and multi-physics systems.
- Learn a systematic way to convert a model of a multi-physics system to an equivalent model of a single physical domain, which is preferably the output domain of the system. For this purpose, learn to use the concepts of energy transfer (or coupling) through generalized transformers and generalized gyrators.
- Gain the ability to relate the learned concepts of modeling to model a mechatronic system. For this purpose, understand the value of a more generalized definition of a mechatronic system.
Design, development, modification, implementation, operation, control, and performance monitoring and evaluation of an engineering system require a sufficient understanding of the system and a suitable ârepresentationâ of the system. In other words, a âmodelâ of the system is required for these practical activities. A model is a convenient representation of the actual system. Properties established and results derived in various âmodel-based approachesâ are associated with the model rather than the actual system, whereas it is to the actual system that the excitations (inputs) are applied to and the responses (outputs) are observed or measured from. This distinction is very important, particularly in the context of the treatment in this book. However, as customary, the terms the model and the system are often used interchangeably to refer to the model. This fact can be easily recognized depending on the specific context that is addressed and is usually not confusing.
An engineering system may consist of several different physical types of components, belonging to such physical âdomainsâ as mechanical, electrical, fluid, and thermal. It is termed a multi-physics (or multi-domain or mixed) system. Furthermore, it may contain multifunctional components; for example, a piezoelectric component, which can interchangeably function as both a sensor and an actuator, is a multifunctional device. It is desirable to use analogous procedures in the modeling of multi-physics and multifunctional components. Then the individual physical-domain models or functional models can be developed using âunifiedâ or âanalogousâ methodologies across the physical domains while considering all the physical domains simultaneously (i.e., âconcurrentâ or âintegratedâ manner), systematically, to obtain a âuniqueâ (i.e., the âbestâ single) overall model.
Analytical models may be developed for mechanical, electrical, fluid, and thermal systems in a rather analogous manner, using the mentioned âunifiedâ approach because clear analogies exist among these four types of systems and in their variables. This is an important focus of this book. In view of the existing analogy, then, a unified (analogous), integrated (concurrent), and systematic (having clear steps) approach may be adopted in the modeling, analysis, design, control, and evaluation of an engineering system. This integrated and unified approach is indeed the âmechatronicâ approach to modeling. The âunifiedâ approach goes beyond the conventional mechatronic procedures (which are âintegratedâ yet may not be unified) and exploits the similarities (analogies) of different physical domains of the system. In summary then, the studies and developments of this book target a modeling approach that has the following characteristics:
- Integrated (concurrent or simultaneous; considers all physical domains of the system simultaneously, while including âcouplingâ or âdynamic interactionsâ or âenergy conversionâ that exist among them)
- Unified (exploits analogies or similarities among different physical domains and uses similar/analogous procedures to model the dynamics in those physical domains)
- Systematic (follows a clearly indicated sequence of modeling steps, without any confusion as to the approach)
- Realization of a âuniqueâ model (the modeling procedure leads to a single âbestâ model). This implicitly implies that some form of âoptimizationâ is associated with the used procedures
- Physically meaningful (e.g., the system variables, particularly the state variables, are not chosen arbitrarily, and have physical meaning, and furthermore, it leads to a clear understanding of the dynamic size or âorderâ of the system).
1.1.1 Model Error of Science Error
A model is a representation of the actual system, with sufficient accuracy. Hence, there bound to be errors in the model, when compared to the real system, but often they can be neglected depending on the purpose of the model. Also, a model represents the physical phenomena in the actual system (again to an acceptable level of accuracy). Such phenomena are never in error, but our understanding and representation/formulation of these phenomena may not be accurate or may be evolving. So, the science itself can have errors, which are corrected from time to time.
1.2 Importance and Applications of Modeling
A dynamic model may be indispensable in a variety of engineering applications. The types of uses of a dynamic model include the following:
- Analysis of a dynamic system (particularly using mathematical methods and tools), even when the actual system is not available or developed yet
- Computer simulation, which can incorporate various types of models including mathematical (analytical) dynamic models and even some physical hardware (i.e., hardware-in-the-loop or SIL simulation)
- Determination of the required design of a dynamic system, prior to building the system (in fact it may assist in making the decision whether to build or not)
- Determination of the required modification of a dynamic system (or its model or the design), prior to the actual task of physical modification of the system
- Instrumentation (i.e., the exercise of âinstrumentingâ) of a dynamic system. Specifically, instruments (such as sensors, actuators, and signal conditioning and component interconnecting hardware) needed for the operation and/or performance improvement of a dynamic system may be established (i.e., selected or sized) and analyzed through modeling and simulation
- Control or assistance in the physical operation of a dynamic system (e.g., for model-based control and for generating control signals and performance specifications)
- Testing of a dynamic system (where a test regiment is developed and evaluated through analytical and computational means) and in product qualification (where an available good-quality product is further tested and evaluated to determine whether it is suitable for a specialized application (e.g., seismic qualification of the components of a nuclear power plant; qualification of computer hardware for shipment))
- Performance evaluation (including online monitoring) of a system to detect deviations and diagnose malfunctions and faults (using a model as the reference for good performance).
Dynamic modeling is applicable in all branches of engineering (aerospace, biomechanical/medical, chemical, civil, electrical and computer, manufacturing, material, mechanical, mechatronic, mining, etc.) and even in non-engineering systems (e.g., social, economic, administrative, environmental). Analytical models are quite useful in predicting the dynamic behavior (response) of a system for various types of excitations (inputs). For example, vibration is a dynamic phenomenon and its analysis, practical utilization, and effective control require a good understanding (model) of the vibrating system. Computer-based studies (e.g., computer simulation) may be carried out using analytical models (sometimes incorporating some physical hardware as wellâhardware in the loop or SIL simulation) while using suitable values for the system parameters (mass, stiffness, damping, capacitance, inductance, resistance, fluid inductance, or inertance and so on). A model may be employed when designing an engineering system for proper performance. Then, the system is first developed (designed) using a model, which is much easier (quick, flexible, inexpensive) to modify than a physical system or prototype. In the context of product testing, for example, analytical models are commonly used to develop test specifications and the input signals that are applied by the exciter in the test procedure. Dynamic effects and interactions in the test object, the excitation system, and their interfaces may be studied in this manner. Product qualification is a particular situation of this, which is the procedure that is used to establish the capability of a good-quality product to withstand a specified set of operating conditions, in a specialized application. In product qualification by testing, the ...
